List of top Mathematics Questions on Product of Matrices

$\text{ If } P = \begin{bmatrix} -1 \\ 2 \\ 1 \end{bmatrix} \text{ and } Q = \begin{bmatrix} 2 & -4 & 1 \end{bmatrix} \text{ are two matrices, then } (PQ)^T \text{ will be:}$

Let $ p = (1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}) \in {R^4 $ and $ f : {R}^4 \to {R} $ be a differentiable function such that $ f(p) = 6 $ and $ f(Ax) = A^3 f(x) $, for every $ A \in (0, \infty) $ and $ x \in {R}^4 $. The value of \[ 12 \frac{\partial f}{\partial x_1}(p) + 6 \frac{\partial f}{\partial x_2}(p) + 4 \frac{\partial f}{\partial x_3}(p) + 3 \frac{\partial f}{\partial x_4}(p) \] is equal to (answer in integer):}

Consider the following Linear Programming Problem $ P $: Minimize $ x_1 + 2x_2 $, subject to
$ 2x_1 + x_2 \leq 2 $,
$ x_1 + x_2 = 1 $,
$ x_1, x_2 \geq 0 $.
The optimal value of the problem $ P $ is equal to:

Let $D = \{(x, y) \in \mathbb{R}^2 : x > 0 \text{ and } y > 0\}$. If the following second-order linear partial differential equation
$y^2 \frac{\partial^2 u}{\partial x^2} - x^2 \frac{\partial^2 u}{\partial y^2} + y \frac{\partial u}{\partial y} = 0$ on $D$
is transformed to
$\left( \frac{\partial^2 u}{\partial \eta^2} - \frac{\partial^2 u}{\partial \xi^2} \right) + \left( \frac{\partial u}{\partial \eta} + \frac{\partial u}{\partial \xi} \right) \frac{1}{2\eta} + \left( \frac{\partial u}{\partial \eta} - \frac{\partial u}{\partial \xi} \right) \frac{1}{2\xi} = 0$ on $D$,
for some $a, b \in \mathbb{R}$, via the coordinate transform $\eta = \frac{x^2}{2}$ and $\xi = \frac{y^2}{2}$, then which one of the following is correct?

Let $ T, S : {R}^4 \to {R}^4 $ be two non-zero, non-identity $ {R} $-linear transformations. Assume $ T^2 = T $. Which of the following is/are true?

Let $ p_1<p_2 $ be the two fixed points of the function $ g(x) = e^x - 2 $, where $ x \in {R} $. For $ x_0 \in {R} $, let the sequence $ (x_n)_{n \geq 1} $ be generated by the fixed-point iteration \[ x_n = g(x_{n-1}), \quad n \geq 1. \] Which one of the following is/are correct?

The problem: Let $ f : \mathbb{R}^2 \to \mathbb{R} $ be a function such that \[ f(x, y) = \begin{cases} \left( 1 - \cos\left(\frac{x^2}{y^2}\right) \right) \sqrt{x^2 + y^2}, & \text{if } y \neq 0, x \in \mathbb{R}, \\ 0, & \text{otherwise.} \end{cases} \] Which of the following is/are correct?

For an integer $ n $, let $ f_n(x) = xe^{-nx }$, where $ x \in [0, 1] $. Let $ S := \{f_n : n \geq 1\} $. Consider the metric space $ (C([0, 1]), d) $, where \[ d(f, g) = \sup_{x \in [0, 1]} |f(x) - g(x)|, \quad f, g \in C([0, 1]). \] Which of the following statement(s) is/are true?}

Let $ T : {R}^4 \to {R}^4 $ be an $ {R} $-linear transformation such that 1 and 2 are the only eigenvalues of $ T $. Suppose the dimensions of ${Kernel}(T - I_4)$ and ${Range}(T - 2I_4)$ are 1 and 2, respectively. Which of the following is/are possible (upper triangular) Jordan canonical form(s) of $ T $?

Let $ L^2([-1, 1]) $ denote the space of all real-valued Lebesgue square-integrable functions on $ [-1, 1] $, with the usual norm $ \|\cdot\| $. Let $ P_1 $ be the subspace of $ L^2([-1, 1]) $ consisting of all the polynomials of degree at most 1. Let $ f \in L^2([-1, 1]) $ be such that \[ \|f\|^2 = \frac{18}{5}, \quad \int_{-1}^1 f(x) dx = 2, \quad {and} \quad \int_{-1}^1 xf(x) dx = 0. \] Then \[ \inf_{g \in P_1} \|f - g\|^2 = ........... \]

The maximum value of $ f(x, y, z) = 10x + 6y - 8z $ subject to the constraints \[ 5x - 2y + 6z \leq 20, \quad 10x + 4y - 6z \leq 30, \quad x, y, z \geq 0, \] is equal to …………. (round off to TWO decimal places).

Let $ K \subseteq \mathbb{C} $ be the field extension of $ \mathbb{Q} $ obtained by adjoining all the roots of the polynomial equation $ (x^2 - 2)(x^2 - 3) = 0 $. The number of distinct fields $ F $ such that $ \mathbb{Q} \subseteq F \subseteq K $ is equal to ……….. (answer in integer).

Let $ r : [0,1] \to \mathbb{R}^2 $ be a continuously differentiable path from $ (0,2) $ to $ (3,0) $ and let \[ \mathbf{F} : \mathbb{R}^2 \to \mathbb{R}^2 be \mathbf{F}(x, y) = (1 - 2y, 1 - 2x). \] The line integral of $ \mathbf{F} $ along $ r $ is equal to $\_\_\_\_\_\_$ (round off to TWO decimal places).

The problem: Let $ u(x,t) $ be the solution of the initial value problem \[ \frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0, \quad x \in \mathbb{R}, \, t > 0, \] \[ u(x,0) = 0, \quad x \in \mathbb{R}, \quad \frac{\partial u}{\partial t}(x,0) = \begin{cases} x^4 (1 - x)^4, & 0 < x < 1, \\ 0, & \text{otherwise}. \end{cases} \] If $ \alpha = \inf \{ t > 0 : u(2,t) > 0 \} $, then $ \alpha $ is equal to $\_\_\_\_\_\_$ (round off to TWO decimal places).

The global maximum of $ f(x, y) = (x^2 + y^2)e^{-x-y} $ on $ {(x, y)} in \mathbb{R}^2 : x \geq 0, y \geq 0\ $ is equal to $\_\_\_\_\_\_$ (round off to TWO decimal places).

Let $ \ell^p = \left\{ x = (x_n)_{n \geq 1 : x_n \in {R}, \|x\|_p = \left( \sum_{n=1}^\infty |x_n|^p \right)^{1/p}<\infty \right\} $ for $ p = 1, 2 $. Let \[ c_0 = \{ (x_n)_{n \geq 1} : x_n = 0 { for all but finitely many } n \geq 1 \}. \] For $ x = (x_n)_{n \geq 1} \in c_0 $, define $ f(x) = \sum_{n=1}^\infty \frac{x_n}{\sqrt{n}} $. Consider the following statements:} 1. There exists a continuous linear functional $ F $ on $ (\ell^1, \|\cdot\|_1) $ such that $ F = f { on } c_0 $.
2. There exists a continuous linear functional $ G $ on $ (\ell^2, \|\cdot\|_2) $ such that $ G = f { on } c_0 $.
Which one of the following is correct?

Let $ \ell^2_{{Z}} = \{ (x_j)_{j \in {Z} : x_j \in {R} { and } \sum_{j = -\infty}^\infty x_j^2<\infty \} $ endowed with the inner product \[ \langle x, y \rangle = \sum_{j = -\infty}^\infty x_j y_j, \quad x = (x_j)_{j \in {Z}}, \, y = (y_j)_{j \in {Z}}. \] Let $ T : \ell^2_{{Z}} \to \ell^2_{{Z}} $ be given by $ T((x_j)_{j \in {Z}}) = (y_j)_{j \in {Z}} $, where \[ y_j = \frac{x_j + x_{-j}}{2}, \quad j \in {Z}. \] Which of the following is/are correct?}

Let $ X $ be the normed space $ ({R}^2, \|\cdot\|) $, where \[ \| (x, y) \| = |x| + |y|, \quad (x, y) \in {R}^2. \] Let $ S = \{ (x, 0) : x \in {R \} $ and $ f : S \to {R} $ be given by $ f((x, 0)) = 2x $ for all $ x \in {R} $. Recall that a Hahn–Banach extension of $ f $ to $ X $ is a continuous linear functional $ F $ on $ X $ such that $ F|_S = f $ and $ \|F\| = \|f\| $, where $ \|F\| $ and $ \|f\| $ are the norms of $ F $ and $ f $ on $ X $ and $ S $, respectively. Which of the following is/are true?}

Let $ g : M_2({R}) \to {R $ be given by $ g(A) = \operatorname{Trace}(A^2) $. Let $ O $ be the $ 2 \times 2 $ zero matrix. The space $ M_2({R}) $ may be identified with $ {R}^4 $ in the usual manner. Which one of the following is correct?}

Consider the following statements:
1. There exists a proper subgroup $ G $ of $ ({Q}, +) $ such that $ {Q}/G $ is a finite group.
2. There exists a subgroup $ G $ of $ ({Q}, +) $ such that $ {Q}/G $ is isomorphic to $ ({Z}, +) $.
Which one of the following is correct?

Let $ (\cdot, \cdot) $ denote the standard inner product on $ {R}^n $. Let $ V = \{v_1, v_2, v_3, v_4, v_5\} \subset {R}^n $ be a set of unit vectors such that $ (v_i, v_j) $ is a non-positive integer for all $ 1 \leq i \neq j \leq 5 $. Define $ N(V) $ to be the number of pairs $ (r, s) $, $ 1 \leq r, s \leq 5 $, such that $ (v_r, v_s) \neq 0 $. The maximum possible value of $ N(V) $ is equal to:

Let $ f(x) = |x| + |x - 1| + |x - 2|, \, x \in [-1, 2] $. Which one of the following numerical integration rules gives the exact value of the integral \[ \int_{-1}^2 f(x) \, dx? \]

Let $ P_f(x) $ be the interpolating polynomial of degree at most two that interpolates the function $ f(x) = x^2|x| $ at the points $ x = -1, 0, 1 $. Then \[ \sup_{x \in [-1, 1] |f(x) - P_f(x)| = \, {(round off to TWO decimal places)}. \] }

The maximum of the function $ f(x, y, z) = xyz $ subject to the constraints \[ xy + yz + zx = 12, \quad x>0, y>0, z>0, \] is equal to (round off to TWO decimal places):

Let $ f : [0, 1] \to {R} $ be a function. Which one of the following is a sufficient condition for $ f $ to be Lebesgue measurable?